If you use dyed-in-the-wool to describe someone or their beliefs, you are saying that they have very strong opinions about something, which they refuse to change.
...a dyed-in-the-wool conservative.
ADJ : ADJ n
DYED-IN-THE-WOOL
Meaning of DYED-IN-THE-WOOL in English
Collins COBUILD Advanced Learner's English Dictionary. Английский словарь Коллинз COBUILD для изучающих язык на продвинутом уровне. 2006